## On an integral inequality

Let and be a continuous and increasing function. Prove that

**Solution**

## An almost zero function

Let be a Riemann integrable function. If forall rationals of the interval then prove that .

**Solution**

## The matrix is symmetric

Let be a square matrix and for that it holds that

Prove that is symmetric.

**Solution**

(1)

Taking transposed matrices back at we get that

and thus

(2)

On the other hand it holds that

since

Of course it holds that if a matrix is symmetric or antisymmetric and then . *The proof is left as an exercise to the reader.*

We can safely conclude using the above observations that for our matrix it holds that

(3)

But then for the matrix it holds that

and since the matrix is antisymmetric we conclude that and thus . Hence the result.

## On permutation

For any permutation define its displacement as

What is greater: the sum of displacements of even permutations or the sum of displacements of odd permutations? The answer may depend on .

**Solution**

## Equal determinants

Let that are diagonizable in . If and then prove that

**Solution**

Hence

where are the diagonial elements of and of . According to we have that

But , so and finally we conclude that: